WASHINGTON — As Mitt Romney tries to translate his well-reviewed debate performance into votes, one sentence he uttered during the Denver matchup provides the key to his strategy: “A woman grabbed my arm and she said, ‘I’ve been out of work since May. Can you help me?’ ”

Romney likely cannot win the general election unless he can convince more women that he understands their concerns and can, indeed, help them. Prior to the debate, the size of the gender gap was one of Romney’s biggest hurdles to becoming president. President Obama led Romney among women by 18 points while trailing among men by 10 points, according to a Quinnipiac University national survey. Moreover, women typically vote in larger numbers than men, amplifying their political power.

Romney’s problem has been clear: Polls show that many women believe that government should do more to help people in need, which has been at odds with Romney’s shrink-the-government message. So, at the debate and afterward, Romney has delivered a retooled and more moderate appeal that is designed to win over key slices of the women’s vote.

“Women are much more concerned than men about preserving a social safety net,” said Susan Carroll, senior scholar at the Center for Women and American Studies at Rutgers University. “That is a problem for Romney,” she said, and explained why he sought to “come across as more compassionate” in the debate.

The fight for the women’s vote escalated in the days after the debate, with Obama using a Friday rally to attack Romney as extreme on an array of social and economic issues concerning women. “I don’t think a working mom . . . should have to wait to get a mammogram because money’s tight,” Obama said, emphasizing that his health care plan covers such tests.

Romney, meanwhile, has ramped up outreach efforts to women, relying heavily on surrogates such as former Massachusetts lieutenant governor Kerry Healey, who served under Romney, to counter Democratic efforts to portray the former Massachusetts governor as uncaring about women’s issues.

“The campaign has been reaching out to women in a number of ways to counteract this specious narrative that has been promoted by the Obama campaign concerning a so-called ‘war on women,’ ” said Healey, who has gone on tours across the country as part of the strategy. “People are always surprised to hear that Governor Romney made a specific effort to have half of his Cabinet be women.”

Romney’s gender gap had been widening in the wake of a secretly recorded video in which Romney said his job wasn’t to worry about 47 percent of Americans who view themselves as “victims” and won’t take personal responsibility. But Obama didn’t bring up the issue in the debate, much to the chagrin of many Democrats. Nor did Obama bring up abortion rights or other social issues of particular concern to women.

While his campaign believes Romney gained support from women as a result of the debate, an array of polling data released in recent weeks reveals the challenge facing him. While the former Massachusetts governor led Obama by one point among married women, he trailed among single women by 37 points, according to the Quinnipiac survey. (About half of the women surveyed were single.)

Republicans once firmly controlled the women’s vote. In 1972, Republican Richard Nixon won the women’s vote by a 60-38 percent margin, according to CBS News exit polls. The women’s vote gradually swung to Democrats, with President Clinton getting a sizable majority in his two campaigns. But in 2004, with terrorism concerns paramount, Senator John F. Kerry of Massachusetts won the women’s vote by only 3 percentage points, a margin so slim that it was a key factor in his defeat. Then, in 2008, Obama won among women by 13 points even though Senator John McCain, the Republican nominee, had hoped his pick of Alaska Governor Sarah Palin as a running mate would bolster his standing.

The gender gap is particularly powerful because women have been going to the polls in greater numbers than men in recent elections. In 2008, 70.4 million women voted, compared with 60.7 million men, according to the Center for Women and American Politics. “Just in terms of sheer numbers women are critically important, but in a close race those votes are decisive,” said the center’s Carroll.

Republicans declared after the last election that they would focus much attention on winning over women. But a series of actions, first on social issues, and then related to Romney’s remarks about Americans who see themselves as “victims,” appear to have hurt him with female voters.

Social issues came to the fore during the Republican primaries, and then were highlighted when conservative radio host Rush Limbaugh called Sandra Fluke a “prostitute” because she was an advocate for insurance coverage of birth control. Romney responded to the Fluke controversy by saying the radio’s host comments were “not the language I would have used,” prompting Democrats to say that the Republican had failed to directly condemn the remarks. Then Todd Akin, a Republican candidate for the US Senate from Missouri, said that in “a legitimate rape, the female body has ways to try to shut that whole thing down,” avoiding pregnancy. Romney responded more forcefully that time, condemning the remarks as “inexcusable” and unsuccessfully urged him to drop out of the race.

That was followed by the adoption of a Republican Party platform plank that called for a ban on all abortions, with no exclusions for rape or incest. (Romney supports exceptions allowing abortion in cases of rape, incest, or life of the mother.) A Republican-friendly columnist, Kathleen Parker, summed up her concerns about the combination of events in a story that Newsweek described on its cover with the headline, “Have Republican Men Gone Crazy?”

Democrats seized on these issues, declaring that Republicans were waging a “war on women.” They used their national convention in Charlotte to try to solidify the women’s vote. Then came the release of the “47 percent” video, which led to a low point in Romney’s standing among women.

Romney’s debate performance was designed to stop the bleeding by addressing concerns raised by women in polls and focus groups. After telling the stories of two women facing economic difficulties, Romney sought to reassure viewers about his view on education. “I’m not going to cut education funding,” Romney said. Months earlier, however, he had said at a private fund-raiser that he would shrink the Department of Education, only mildly endorsing the agency by saying that “I’m not going to get rid of it entirely.”

Romney also softened his characterization of health care, arguing that his program would provide affordable insurance for families, instead of just emphasizing his opposition to the president’s plan.

Debbie Wasserman Schultz, chairwoman of the Democratic National Committee, said in a telephone interview that Romney “lied” in an effort to mislead women about his positions. She said, for example, that Romney’s statement that his plan would insure people with preexisting conditions is not true. Romney has said the requirement to cover people with preexisting conditions only applies to those who have had continuous insurance.

Schultz said the DNC is overseeing a massive effort to explain Obama’s positions to female voters, including “women to women phone banks,” organizational efforts, and female-specific advertising. For example, one ad portrays Romney as “extreme” because he doesn’t “understand the mindset of someone who has to go to Planned Parenthood.”

Courtney Johnson, the Romney campaign’s deputy national coalitions director, who has attended dozens of meetings with women and campaign supporters, said Romney knows the primary concern of women is the economy.

“Women are too smart to base the decision on something as important as the future direction of this country and our economy on social issues alone,” Johnson said. “We are out there talking to women and listening to them to hear about the issues that are important to themselves and their families.”

Peter Brown, the director of the Quinnipiac poll, said the economy, not social issues, is the major concern of female voters.

“Everybody wants to make this about reproductive rights,” Brown said. “It isn’t. This is about money. This is about the role of government. Women are more supportive of the role of government than men. Women are more likely to be more reticent about cuts in government programs than men. Men and women are different. I know that’s a revelation but that’s really what it boils down to.”

The key to Romney’s strategy could be whether he can win over blue-collar women. In New Hampshire, for example, Romney had been ahead among working-class women by one percentage point in early September.

A few weeks later, however, Romney trailed Obama by 15 points among that group, according to Andy Smith, a University of New Hampshire pollster who conducted a survey for WMUR-TV.

Smith said that the shift — the largest such demographic movement he has seen in this election — may be the result of publicity about Romney’s “47 percent” comment.

Romney last week sought to distance himself from his own remark, saying on Fox News that he had been “completely wrong.”

“When I become president, it will be about helping the 100 percent,” Romney said.